On relative normal modes

C. R. Acad. Sci. Paris, t. 328, SCrie I, p. 413-418, CComGtrie diff~rentielle/Di#erentia/ Geometry (Systtimes dynamiques/Dynamica/ Systems)

On relative

normal

Eugene

Tadashi

LERMAN,

Department of Mathematics, E-mail: [email protected], (Requ

le 23

novembre

Abstract.

modes TOKIEDA

University of Illinois, [email protected] 1998,

accept6

le 7 dBcembre

Urbana

IL

61801,

USA

1998)

We generalizethe Weinstein-Mosertheorem on the existenceof nonlinear normal modesnear an equilibrium in a Hamiltoniansystemto a theoremon the existence of relative periodic orbits near a relative equilibrium in a Hamiltonian systemwith continuoussymmetries.In particular,we prove that underappropriatehypothesesthere exist relative periodic orbitsnearrelative equilibriaeven when theserelative equilibria are singularpoints of the correspondingmomentmap, i.e. when the reducedspaces are singular.0 AcadCmiedesSciences/Elsevier,Paris SW les oscillations

R&urn&

1999

norm&es

relatives

On gt!nne’ralise le thtforkme d’existence de Weinstein-Moser des oscillations normales non linkaires voisines d’un kquilibre dans un systkme hamiltonien, ci un th.4ortime d’existence des orbites pe’riodiques relatives voisines d’un kquilibre relatif dans un systkme hamiltonien ci syme’trie continue. Entre autres on dkmontre, moyennant quelques hypothlses, qu’il existe des orbites p&iodiques relatives prks des kquilibres relatifs mCme lorsque ceux-ci sont des points singuliers de l’application de moment, c’est-irdire lorsque les espaces rkduits sont singuliers. 0 Acadkmie des ScienceslElsevier,

Paris

Version franqaise

abrdghe

Dans cette Note on gbn&alise le thCor&me de Weinstein-Moser [13], [9], [14], [8], [2] sur les oscillations normales non linkaires voisines d’un 6quilibre dans un sysdme hamiltonien, g un thbortime

sur les orbites pkriodiques relatives voisines d’un kquilibre relatif dans un systkme hamiltonien symktrique. Soit A4 une variCt& symplectique munie d’une opkation hamiltonienne d’un groupe de Lie connexe G, et done d’une application de moment Cp : M --+ II*. Une orbite du champ hamiltonien Xh d’un hamiltonien G-invariant h E C”( M)G est un kquilibre relatif (resp. une orbite pe’riodique relative) si son image dans M/G est un point (resp. un lacet). Les amkliorations ulttkieures du thCor2me de Weinstein-Moser se p&tent toutes ?Ides gCnCralisations par notre mtthode; ntanmoins, afin de ne pas alourdir l’expod on traitera seulement la version originale : Note prksentke par Charles-MichelMARLE. 0764~4442/99/032804

13 0 Academic

des ScienceslElsevier,

Paris

413

E. Lerman,

T. Tokieda

THI?OR~ME DE WEINSTEN [ 131. - Soit h un hamiltonien sur un vectoriel symplectique V avec dh(0) nul et d2h( 0) dkjini positit Alors pour tout & > 0 petit, h-l (h(0) + E) Porte au moins i dim V orbites pe’riodiques de Xh.

Soit z E M un point d’un Cquilibre relatif pour un hamiltonien G-invariant. Existe-t-i1 des orbites pkriodiques relatives voisines ? A cela nulle difficult6 si :c est un point rkgulier de a, ou meme dans le cas singulier si la dimension de la strate de son image rCdans l’espace riduit est strictement positive, car le probl&me se ramkne alors au thCo&me de Weinstein sur cette strate, qui est stable par rapport & la dynamique de l’espace rkduit stratifiC [1], [12]. Mais que dire si la strate de ?Zest un point ? On verra qu’en l’occurrence il existe encore des familles d’orbites pkriodiques relatives voisines de I’Cquilibre relatif, pourvu qu’un certain <du hessien soit dCfini en tant que forme quadratique et que le groupe d’isotropie de O(x) soit un tore. On a en effet un thCo&me de correspondance que voici : TH~ORI~ME 1. - Soit M une varie’te’symplectique munie d’une ope’ration hamiltonienne d’un groupe

de Lie connexe G et d’une application de moment Q, : M --f g*, et soit h E C-(M)G un hamiltonien G-invariant. Supposonsque x E M appartient & un kquilibre relatifpour h et que le groupe d’isotropie de (a(z) soit un tore. Alors il existe un vectoriel symplectique V muni d’une ope’ration hamiltonienne du groupe d’isotropie G, de x et un hamiltonien G,-invariant hv E C-(V)Gx tels que : 1. l’origine 0 E V est un kquilibre pour hlr ; 2. le hessien d2hv(0) de hr. peut e^trecalcule’ h partir de h ; 3. toute orbite pe’riodique G,-relative pour hl/ duns V donne lieu i2unefamille d’orbites pe’riodiques G-relatives pour h duns M. Ici V dCsigne le sous-espacesymplectique maximal de ker d+(x) (dit tranche symplectique en 2) et d2hr/(0) est la forme quadratique induite sur V par d2(h - (@l<))(~)l~~~,i+(~), oti < est une vitesse de x que l’on trouve dans : x appartient & un Cquilibre relatif pour h +=+ 31 E g,

d(h - (@l<))(x) = 0.

La forme normale locale de Guillemin-Sternberg et de Marle [4], [7] permet de construire une fonction ht- sur V dont le hessien est d2hv(0). Cette notion de hessien intervient dans I’Ctude de la stabilitC et des bifurcations des Cquilibres relatifs aux points singuliers de l’application de moment [6], [lo], [I 11. Le thCor?me 1, joint par exemple au thCo&me de Weinstein, conduit au resultat escomptC: COROLLAIRE. - Soient M, G, @ : M -+ g*, h E C-(M)G comme duns le the’orsme 1. Supposons que x E M appartient ri un e’quilibre relatif et que le groupe d’isotropie de @p(x) est un tore; appelons V la tranche symplectique en x. S’il existe une vitesse < de x telle que d2( h - (@I,$))(x) IIT soit dt;Jini positij alors pour tout E > 0 petit, {y E Ml(h - (@~[))(zJ) = (h - ((alE))(x) + &} Porte des familles d ‘orbites phiodiques relatives.

On peut saris doute se passer de l’hypoth&e selon laquelle le groupe d’isotropie de @(cc) est un tore, mais la dkmonstration ne promet g&-e d’etre si simple. Signalons que dans le cas oti G est compact, l’hypothkse est satisfaite de faGon g&z&ique. Esquissede la dkmonstration. - D’abord, on se ramkne au cas oti G est un tore en passant5 un soussystbmehamiltonien dont le groupe de symCtrie est le groupe d’isotropie de a(x), gr&e h un thCor&me de Guillemin-Sternberg (voir [3], corollaire 2.3.6). Ensuite, on plonge, par un symplectomorphisme equivariant, un voisinage de G .x dans (T* L x V)/lY (L est le tore ComplCmentairede la composante connexe K de 1 dans G,, et r = G,/K) ; en rkduisant par L, on rkduit ce dernier espace B V

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et h h hv. Enfin, on verifie que les orbites periodiques relatives pour h\r se relevent a des orbites periodiques relatives pour h : les unes et les autres correspondent aux orbites periodiques du systeme obtenu soit en reduisant V par l? x K, soit en reduisant (T*L x V)/r par L x K. Comme un exemple d’application, on remarque que le corollaire assure l’existence des orbites periodiques relatives d’une paire de corps rigides a symetrie axiale lies par un potentiel qui depend de l’angle entre les corps.

1. Introduction In this paper we generalize the Weinstein-Moser Theorem ([13], [9], [14], [8], [2], and references therein) on the nonlinear normal modes near an equilibrium in a Hamiltonian system to a theorem on the relative periodic orbits near a relative equilibrium in a symmetric Hamiltonian system. Let A4 be a symplectic manifold, with a Hamiltonian action of a connected Lie group G and a moment map + : M -+ g*. Recall that an orbit of the Hamiltonian vector field Xh of a G-invariant Hamiltonian h E C”(M) G is a relative equilibrium if its image in the orbit space M/G is a point, and that an orbit of Xl, is a relative periodic orbit if its image in AJ/G is a closed curve. Later improvements of the Weinstein-Moser theorem lend themselves to generalizations by our method; to streamline the presentation, however, we shall treat only the original version: WEINSTEIN’S THEOREM [ 131. - Let h be a Hamiltonian on a symplectic vector space V such that its differential at the origin dh(0) zs . zero and its Hessian at the origin d’h,(O) is positive dejinite. Then for every small E > 0, the energy level h-‘(h(0) + E) carries at least i dim V periodic orbits of the Hamiltonian vector held of h,.

Now suppose 2 E iM lies on a relative equilibrium for a G-invariant Hamiltonian h. If 2: is a regular point of the moment map Cp,then the reduced space at h= a(z) is smooth near the image 3: of 2. Provided appropriate conditions hold on the Hessian of the reduced Hamiltonian, Weinstein’s theorem applied to the reduced system guaranteesat least $ dim (reduced space) periodic orbits, and hence as many Familiesof relative periodic orbits near the relative equilibrium. (If zz lies on a relative periodic orbit, then the orbit through g . 5‘ is relative periodic also for every .g E G.) On the other hand, if z is a singular point of @, then the reduced space at p is a stratified space, and the reduced dynamics preserves the stratification [ 11, [12]. Unless the stratum through : is an isolated point, we have again i dim (stratum) families of relative periodic orbits, provided appropriate conditions hold on the Hessian of the restriction of the reduced Hamiltonian to the stratum. But what if the stratum through Z is an isolated point? It is difficult to make senseof Hessianson singular spaces. We shall show that in this case also there are families of relative periodic orbits near the relative equilibrium provided a certain substitute for the Hessian is definite and the isotropy group G, of p is a torus. The proof amounts to a computation in “good coordinates” that allows us to reduce our problem to Weinstein’s theorem. We proceed via the following “correspondence theorem”: THEOREM 1. - Let M be a symplectic manifold, with a Hamiltonian action of a connected Lie group G and a moment map Q : M + g*, and let h E C-(M)G be a G-invariant Hamiltonian. Supposex E .‘II lies on a relative equilibrium for h and the isotropy group of@(x) is a torus, Then there exist a symplectic vector space V with a Hamiltonian action qf the isotropy group G,. of x and a G,.-invariant Hamiltonian h\r E C-(V)Gz such that: 1. the origin 0 E V is an equilibrium for hlJ; 2. the Hessian d’hV(O) of h v can be computed in terms of h; 3. every G,-relative periodic orbit for hv on V sufficiently close to the origin gives rise to a family of G-relative periodic orbits for h on M.

415

E. Lerman,

T. Tokieda

The meaning of the vector space V and of the quadratic form d’hv(0) 5 E iU lies on a relative equilibrium

for h w

is as follows. Note that

d( h - (@I[)) (x) = 0 for some < E g.

The vector < is called a velocity of 2. Velocity is not unique: any two velocities of z differ by a vector in the isotropy Lie algebra gz of 1~. The function h - (@I<) is constant along the orbit G, . 5, where as above G,, is the isotropy group is therefore always degenerate and descends to of P = a(x). The form d2(h - (@it))(~)lkerd@(z) a well-defined form on the vector space V := ker d@(z)/T,(G, . z); alternatively, we can think of V as the maximal symplectic subspace of ker d@(Z). V is called the symplectic s&x at .7:. It follows from the local normal form theorem of Guillemin-Sternberg and Marle [4], [7] that there exists a G,-invariant symplectic submanifold C passing through z such that T,C = V. Thus, locally C II V as symplectic manifolds, with z corresponding to the origin in V. The function hsin Theorem 1 is the restriction (h - (@I<)) 1~ = (h - (@I<)) 1~. Since Hessians are functorial, d2hv(0) = d2(h - (Ql<))l~~(O) = d2(h - (@,l<))(z)l\:. ‘Th’1s notion of Hessian has been used in [6], [lo], [ 1 l] to study the stability and bifurcation of relative equilibria at singular points of the moment map. As an example of applications of Theorem 1, we combine it with Weinstein’s theorem of obtain: COROLLARY.-L&M, G, a: M + g*, h E C”( M)G be as in Theorem 1. SupposeII: E M lies on a relative equilibrium for h, and the isotropy group of G(x) is a torus; call V the symplectic slice at z. If there is a velocity < of x for which d2( h - (@I<))(x) 117is positive definite, then for every small E > 0, the level set {y E Mj(h- ((al<))(y) = (h- (@I<))(x)+ E,‘-1~7carries families of relative periodic orbits.

We expect that the assumption on the isotropy group of Q(z) being a torus can be dropped, but the proof is unlikely to be quite so simple. In case G is compact, the assumption is satisfied generically. 2. Proof of Theorem

1

Let AJ, G, CI,: M + g*, h E C-(M)G be as in the statement of Theorem 1. We are supposing that x E M lies on a relative equilibrium for h and that the isotropy group G,, of IL = a(z) is a torus. First, we show that it suffices to consider the case when the whole G is a torus. Since G, is compact, we can produce an Ad+ (G,)-invariant inner product on g* and take the corresponding G,,-equivariant splitting g = gfi $ B. Then a small enough G,-invariant neighborhood B of p in the affine plane PUS$O is transverse to the moment map (here b” denotes the annihilator of b in 8). Hence R := @-l(B) is a G,-invariant submanifold of M containing 2. In fact, by the symplectic cross-section theorem of Guillemin and Sternberg (cJ: [3], Corollary 2.3.6), R is a symplectic submanifold of Al and the action of G, on R is Hamiltonian; the moment map for this action is the restriction of Q, to R followed by the natural projection g* -+ g;. Since g* + g; restricted to I-L+ b” is an isomorphism, the moment map for the action of G,, on R is @‘JR up to a linear isomorphism. It follows that kerd@(y) = kerd(iBlR)(y) for every y E R (c$ [6], Lemma 2.5). Moreover, because h is G-invariant, the flow of Xh preserves the fibers of the moment map, and so the flow preserves R. It follows that (Xh)lIz = XVIR) (c. [SJ, p. 218). Thus, we have found a Hamiltonian sub-system (R, G,, +lR, hlR) for which the symmetric* group G, is a torus. Passing to this sub-system, we may and shall assume without loss of generality that G is a torus and G = G,,. 416

On relative

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Second, we construct the Hamiltonian hv. The connected component K of 1 in the isotropy group G, of z is a torus and has a complementary torus L, so that G = L x K. The finite abelian group r = G,/K may be regarded as a subgroup of L. Then r acts symplectically (by the lifted action) on T*L = L x t* and on the symplectic slice V at :I:. Hence I‘ acts diagonally on T* L x V. Note that G too acts on T* L x V : L acts on its own cotangent bundle and K acts on V. Hence G acts in a Hamiltonian fashion on (T*L x V)/I’. The orbit of [l, O,O] E (T*L x V)/I’ is isotropic and is isomorphic to G/(I’ x K) N G X. By the equivariant isotropic embedding theorem, there exist G-invariant neighborhoods Ue of [l, O,O] in (T*L x V)/I’ and U of z in M and an equivariant symplectomorphism 0 : Uo -+ Ii such that n([l, O,O]) = .‘I and the derivative d41,0~01) sendsV c T~,II,oI( T*L x V)/I’ to V c kerd@(z) c T,rM by the identity map. Define ii = o*(h - ((al()). BecauseG is assumedto be abelian, c is trivially in the center of 0, and so x is G-invariant. At this juncture it is convenient to pretend that Ua = (T*L x V)/I’. By the invariance of lh E C”((T*L X T/)/f?) LxK, ?;([& A, w]) = z([l, X, ?I]) for all (!, X, II) E L x I” x V. Now define hc~(w) = ?;.([l:O,v]). This hlr E Cm(V)rxK is the desired Hamiltonian: we have dhlr(0) = &([l,O>O])J1~ = 0 and d2h\,(0) = d2h([l,0,0])]\;

= d2(a*(h - (~lI)))([l,O,O])I~;

= g*(d% - (+]~))(x)]\-).

Third and last, we prove that relative periodic orbits for hl~ give rise to relative periodic orbits for h. Consider the fully reduced system (P, hp) obtained by reducing the system (V!>IT) by the action of I? x K. We can arrive at (E’: hp) also by reducing the system ((T*L x V)/I’, h) by the action of G = L x K. Thus, relative periodic orb@ for hi, correspond to periodic orbits for hp, which in turn give rise to relative periodic orbits for h, or equivalently to relative periodic orbits for h - (@I[). But relative periodic orbits for h - (@I<) are relative periodic orbits for h. Indeed, writing ‘p; for the Hamiltonian flow of f, we have ‘pja-(Q,c) = ~t(~,~) o q$, by the G-invariance of h. Therefore, ‘pL-(+,o (x) = g . .7: for some g E G if and only if (pi (:I:) =: 9’ . :I: with 9’ = exp(-t<)g E G. This concludes the proof of Theorem 1.

3. Concluding

remarks

1. Corollary proves the existence of relative periodic orbits for a pair of axially symmetric rigid bodies subject to a coupling potential which dependson the angle between the bodies. We expect the theorem to prove the existence of relative periodic orbits for the motion of Riemann ellipsoids. 2. In Theorem 1, we compute d2hv(0) from d2(h - (+]<))(~)]k~~d+(.~) by “killing” the direction of degeneracy. This computation needs only the existence of the “Darboux coordinates” CT: (T*L x V)/l’ > iYO-+ 111(see the proof above), but by computing without the explicit form of (T, we lose some information. Alternatively we can compute the Taylor expansion of (T and translate the higher-order information on the jet of the original Hamiltonian h into the information on the jet of the model Hamiltonian hL7.

References [I] Arms J., Marsden .I., Moncrief V., Symmetry and bifurcations of momentum mappings, Commun. Math. Phys. 78 (4) (1980181) 455-478. [2] Bartsch T., A generalization of the Weinstein-Moser theorems on periodic orbits of a Hamiltonian system near an equilibrium, Ann. Inst. H. Poincark Anal. non Lintaire 14 (6) (1997) 691-7 18.

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[3] Guillemin V., Lerman E., Stemberg S., Symplectic fibrations and multiplicity diagrams, Cambridge University Press, Cambridge, 1996. [4] Guillemin V., Stemberg S., A normal form for the moment map, in: Differential Geometric Methods in Mathematical Physics, S. Sternberg (Ed.), Reidel, Dordrecht, 1984. [5] Lerman E., On the centralizer of invariant functions on a Hamiltonian G-space, J. Differ. Geom. 30 (1989) 805-815. [6] Lerman E., Singer S.F.. Stability and persistence of relative equilibria at singular values of the moment map, Nonlinearity 11 (1998) l-13. [7] Marle C.-M., Modtle d’action hamiltonienne d’un groupe de Lie sur une variete symplectique, Rend. Sem. Mat. 43 (1985) 227-2.51. [8] Montaldi J.A., Roberts R.M., Stewart I.N., Periodic solutions near equilibria of symmetric Hamiltonian systems, Philos. Trans. Roy. Sot. London A 325 (1988) 237-293. [9] Moser J., Periodic orbits near an equilibrium and a theorem of A. Weinstein, Pure Appl. Math. 29 (1976) 727-747. [IO] Ortega J.-P.. Symmetry, reduction and stability in Hamiltonian systems, thesis, University of California, Santa Cruz, 1998. [I l] Ortega J.-P., Ratiu T., Stability of Hamiltonian relative equilibria, preprint, 1997. [ 121 Sjamaar R., Lerman E., Stratified symplectic spaces and reduction, Ann. Math. 134 (1991) 375-422. [13] Weinstein A., Normal modes for nonlinear Hamiltonian systems. Invent. Math. 20 (1973) 47-57. 1141 Weinstein A., Bifurcations and Hamilton’s principle, Math. Z. 159 (1978) 235-248.

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